"It just likes a movie’s preview" - It is just like a movie preview
"For being an active reader, before I read I have to look around the title, pictures, and captions, first and last sentence" - To be an active reader, I have to look at the title, pictures, captions and the first and last sentence before I even begin to read.
"It helped me to create a “big picture” of the article that made me understand articles more easily" - This creates a 'big picture' of the article that helps me to understand it better
"the article-A New Dawn" - the article "A New Dawn"
"It was about a young Cuban Lester Moreno Perez’s escape story" - drop the 'a'
if it was 'a young Cuban's escape story' the 'a' would be fine as it would be modifying 'Cuban', but since you put the name in there, you wouldn't say 'a Lester Moreno Perez'
"Even though I didn’t agree with Lester’s decisions, but I still learned something from him, about a person who wants to succeed they have to be persistent. That’s what I learned from the article and how I improved by the reading."
'even though' and 'but' don't go together, use one or the other, but not both
Even though I didn't agree with Lester's decisions, I still learned something from him. If a person wants to succeed, they have to be persistent.